Consider GWs emitted at two different times \(t_e\)  and \(t_e^{\prime }\) , with wavenumber \(k\)  and \(k^{\prime }\) , and received at corresponding arrival times \(t_0\)  and \(t_0^{\prime }\) , then the different velocities of modes with different frequencies lead to difference of their arrival times [1]}, [2]}
\(\Delta t_0 &=& (1+z) \Delta t_e + \frac{1}{2} \left(\frac{k^{\beta _{\bar{\mu }}}}{M_{\rm LV}^{\beta _{\bar{\mu }}}} -\frac{k^{\prime \beta _{\bar{\mu }}}}{M_{\rm LV}^{\beta _{\bar{\mu }}}} \right)\int _{t_e}^{t_0} \frac{\alpha _{\bar{\mu }}}{a^{\beta _{\bar{\mu }}+1}} dt\nonumber \\&=&(1+z)\Delta t_e + \frac{k^2-k^{\prime 2}}{2} \int _{t_e}^{t_0} \frac{{\cal W}_2 - {\cal G}_2}{a^{3}}dt.\) 
